Feedforward controller for synchronous reluctance machines

ABSTRACT

The present invention provides an electro-mechanical energy exchange system with a variable speed synchronous reluctance motor-generator having an all-metal rotor. A bi-directional AC-to-DC electric power converter interconnects the motor-generator with a DC bus. First and second hybrid controllers provide current regulation for the motor-generator and voltage regulation for the DC bus. Use of both feedback and feedforward control elements provides a controller particularly suited for operating high speed devices.

This application is a continuation of U.S. application Ser. No.10/887,344 filed Jul. 7, 2004, which claims priority from andincorporates Provisional Application 60/484,674 filed Jul. 7, 2003.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to the electro-mechanical arts and energystorage systems. In particular, the present invention pertains tomechanical energy exchange systems coupled with electrical energyexchange systems.

2. Description of Related Art

Electro-mechanical energy exchange systems have provided mechanical andelectrical power solutions for over one hundred years. These solutionshave typically involved a prime mover driving an AC generator at a fixedspeed multiple of the synchronous frequency. These power solutions havenot required electronic processing of the generator output since thegenerator is a constant speed machine able to generate a sinusoidalelectric output at the desired fixed frequency.

Advanced mechanical energy storage devices like high speed flywheelspose new challenges to traditional electro-mechanical energy exchangesolutions. No longer able to rely on fixed speed operation and theattendant fixed frequency of a connected AC generator, these new systemsrequire that each watt of electric power produced in a variable speedgenerator be processed through power electronics using semiconductorswitches to synthesize a fixed frequency AC output.

With the need to process variable frequency AC power using powerelectronics comes the need for high speed semiconductor switchingdevices. At high shaft speeds and hence high electrical frequencies, theresolution of command voltages used to switch the semiconductors on andoff decreases due to a fixed semiconductor switching frequency. Thiscreates difficulties with feedback control techniques typically used tocontrol these systems since the assumptions of continuous-time controltheory typically used to develop feedback controllers become lessappropriate.

SUMMARY OF THE INVENTION

Now, in accordance with the invention, there has been found asynchronous reluctance machine and control system including abi-directional AC-to-DC electric power converter interconnecting andexchanging electric power between a synchronous reluctancemotor-generator and a DC bus wherein said power exchange is controlledby a plurality of controllers operably coupled to said converter andwherein at least one of the controllers is a feedforward controller.

Further, there has been found an energy conversion system comprising abi-directional AC-to-DC electric power converter interconnecting andexchanging electric power between a synchronous reluctancemotor-generator having an all-metal rotor rotatably coupled to amechanical energy exchange device like a flywheel and a DC bus. Thepower exchange is controlled by a plurality of current controllersoperably coupled to said converter wherein a first controller is afeedforward controller and a second controller is a feedback controller.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention is described with reference to the accompanyingdrawings that illustrate the present invention and, together with thedescription, explain the principles of the invention enabling a personskilled in the relevant art to make and use the invention.

FIG. 1 is a diagram showing modules included in the feedforwardcontroller for synchronous reluctance machines constructed in accordancewith the present invention.

FIG. 2 is a diagram showing elements of a second hybrid controller ofthe feedforward controller for synchronous reluctance machines of FIG.1.

FIG. 3. is a diagram showing feedback control elements of a first hybridcontroller of the feedforward controller for synchronous reluctancemachines of FIG. 1.

FIG. 4 is a diagram showing feedforward elements of a first hybridcontroller of the feedforward controller for synchronous reluctancemachines of FIG. 1.

FIG. 5 is a diagram showing elements of the bidirectional AC-to-DCelectric power converter of the feedforward controller for synchronousreluctance machines of FIG. 1.

FIG. 6 is a chart showing operating modes of the second hybridcontroller of the feedforward controller for synchronous reluctancemachines of FIG. 1.

FIG. 7 is a chart showing operating modes of the first hybrid controllerof the feedforward controller for synchronous reluctance machines ofFIG. 1.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

FIG. 1 shows the feedforward controller for a synchronous reluctancemachine 100 of the present invention. The feedforward controller for asynchronous reluctance machine includes synchronous reluctance machinemodule 184, bi-directional AC-to-DC electric power converter 136, firsthybrid controller 180, and second hybrid controller 182.

The machine module 184 includes a synchronous reluctance motor-generator102. The motor-generator includes a rotor 107 having a plurality ofrotor lobes 114 and an electrical stator 118 spaced apart from the rotorby an air gap 174. The rotor 107 may be an all-metal rotor formedentirely from electrically conductive materials. The rotor is integralwith a first shaft portion 112. The first shaft portion has a shaftcoupling 110 that is connected to a mechanical energy exchange device104 and rotates at an angular velocity wre. A shaft speed transducer 116is proximate to the first shaft portion. A rotor position signalconductor 120 interconnects the transducer and a third controller 124.The third controller outputs include the shaft angular position output126 and the shaft angular speed wre signal output 128. The speed sensoris selected from devices employing a known technology including magneticand or optical sensing technologies.

As a person of ordinary skill in the art will recognize, the mechanicalenergy exchange device 104 may be a single device or multipleinterconnected devices. Mechanical energy exchange devices includeflywheels, prime movers, electric motors, non-electric motors, and otherdevices having a rotatable mechanical connection. Optional flywheel mass106 is shown coupled to the first shaft portion 112 by a second shaftportion 108.

The converter 136 interconnects motor-generator 102 with a DC bus 146.The converter includes a driver module 121. The electrical phases a, b,c of the motor-generator are connected to respective converter AC inputs140, 142, 144. First and second converter DC outputs 146, 147 areconnected to respective first and second DC bus conductors 188, 190.Phase current signals ia and ib are provided at the respective outputs168, 172 of the respective first and second phase current sensors 166,170.

The DC bus interconnects the converter 136 with an electrical network138 via first and second DC bus conductors 188, 190. A capacitor 186 isconnected in parallel with the DC bus. The capacitor may be a singledevice or multiple interconnected devices and it may be a film,electrolytic, or super capacitor type or another known electrical devicehaving electrical energy storage capabilities. Bus voltage signal Vbusis provided at the output 149 of parallel connected DC bus voltagesensor 153. The bus current signal Ibus is provided at the output 151 ofthe series connected DC bus current sensor 150.

The first hybrid controller 180 interconnects the converter 136 and thesecond hybrid controller 182. The first hybrid controller includes firsthybrid controller signal input block 103 and first hybrid controllermodule 105. Control voltage bus 178 interconnects first hybridcontroller output 176 with converter input 123. The signal input block103 is interconnected with ia, ib, position, and Vbus signals viarespective signal conductors.

The second hybrid controller 182 includes second hybrid controllersignal input block 113 and second hybrid controller module 115. A peakcurrent signal conductor 184 interconnects a peak current output 111 ofthe second hybrid controller 182 with input block 103 of the firsthybrid controller 180. The second hybrid controller signal input block113 is interconnected with wre, Vreg, Icharge, Zcharge, Vcharge, Vbus,and Ibus signals via respective signal conductors.

FIG. 2 shows details of second hybrid controller 182. The second hybridcontroller includes a first controller 202 that is a feedback busvoltage regulator. The second hybrid controller also includes a secondcontroller 204 that is a charge current regulator. Respective first andsecond controller outputs Ipeakb and Ipeakc are inputs to switch S5.Switch S5 provides for the selection of either Ipeakb or Ipeakc as itsoutput Ipeak.

The first controller 202 provides a current command output Ipeakbcalculated to reduce the error between bus voltage Vbus and a regulationvoltage Vreg. The first controller's output Ipeakb is ((Vbus *Isum)/wr)as implemented in the mathblock1 210. Isum is (Ibus+Irestore) asimplemented in the mathblock2 212 where Irestore is the output of firstproportional integral (PI) controller 208. The error signal error1 inputto controller 208 is the difference between inputs (Vreg−Vbus) asimplemented in the mathblock3 206. The rotor velocity wr is (wre/(2/P)as implemented in the mathblock4 214. P is the number of poles of thesynchronous reluctance motor-generator 102.

The second controller 204 provides a current command output Ipeakccalculated to reduce the error between bus current Ibus and a regulationcurrent Ireg. The second 216 controller's output Ipeakc is the output ofa second proportional integral (PI) controller 216. The error signalerror3 input to the controller 216 is the lesser of the difference(Ibus-Ireg), as implemented in the mathblock5 218, and Ichargemax asimplemented in the limiter 224. Ibus and Ichargemax are controllerinputs. Ireg is the product (zcharge×error2) as implemented in themathblock6 220. zcharge is a controller input. Error signal error2 is(Vbus−Vcharge) as implemented in the mathblock7 222.

FIGS. 3 and 4 show elements of the first hybrid controller 180. FIG. 3shows a third controller 300 that is a feedback controller element andFIG. 4 shows a fourth controller 400 that is a feedforward controllerelement. The vq, vd control voltage bus 178 is connected to either theoutput of the third controller or the output of the fourth controllervia switches S3 and S4 respectively.

FIG. 3 shows third controller 300 that operates to reduce error terms(Idr-idr) and (Idq-idq). Idr and Iqr are control currents derived fromthe Ipeak signal output of the second hybrid controller 182. Currentsidr and idq are feedback signals derived from currents measured inphases a and b of the motor-generator.

-   Idr=Ipeak(Kd)-   Iqr=Ipeak(Kq)-   Kd=f(wre)-   Kq=sqrt(1-(Kd{circumflex over (0)}2))-   The third controller 300 includes a third proportional integral (PI)    synchronous controller 302, a rotor frame transformation block 304,    an inverse rotor frame transformation block 306, input switch S2,    output switches S3, mathblock8 308, mathblock9 310, and mathblock10    312.

The output vd, vq of third controller 300 is provided by the output ofswitch S3 via control voltage bus 178 to the driver module 121 whenswitch S3 is closed. The output of the inverse rotor frame transformblock 306 provides the vd, vq inputs to switch S3. The third PIcontroller 302 outputs provide the rotor reference frame direct andquadrature voltages vdr, vqr to the inputs of the inverse transformblock 306. The Idr input to controller 302 is (Ipeak×Kd) as implementedin the mathblock8 308 where Kd is a function of wre. The Iqr input tocontroller 302 is (Ipeak×Kq) as implemented in the mathblock9 310 whereKq is a function of wre. The rotor frame transform block 304 providesmeasured current signals in the rotor reference frame idr, idq as inputsto controller 302. Inputs to the rotor frame transform block 304 includemeasured current signals id, iq. The id input is equal to ia. The iqsignal is a function of ia, ib as implemented in the mathblock10 312(iq=ia(q/sqrt(3))+ib(2/sqrt(3))).

An additional input to the rotor frame and inverse rotor frame transformblocks 304, 306 is a position signal provided by the output of switchS4. Inputs to switch S4 include Start-Up Angle, Zero, and Position.

FIG. 4 shows fourth controller 400. This controller provides direct andquadrature voltage output commands vd, vq based on a predictive model ofsynchronous reluctance motor-generator 102. The direct and quadraturecontrol voltages each depend upon resistive voltage drop, inductive backemf, and air gap flux back emf terms.

The control voltage output vd, vq of the controller 400 is provided bythe output of switch S4 via control voltage bus 178 to the driver module121 when switch S4 is closed. The vd, vq inputs to switch S4 are theoutputs of inverse transform block 402. the vdr input to the inversetransform block is (R(idr)-Llq(wre)iqr-wre(λqr)) as implemented in themathblock11 406. Mathblock12 410 implements R(idr), the direct value ofresistive voltage drop in the stator 118. Mathblock13 412 implementsLlq(wre)iqr, the direct inductive back electromotive force in thestator. Mathblock14 414 implements wre(λqr), the direct air gap fluxback electromotive force resulting from the airgap 174 between thestator 118 and the rotor lobes 114. The direct air gap flux is evaluatedas follows:${\lambda\quad{dr}} = {{\int_{0}^{t}{{- \frac{R{\mathbb{d}r}}{L{\mathbb{d}r}}}\quad\lambda{\mathbb{d}r}}} + {R{\mathbb{d}{r\left( \frac{M\mathbb{d}}{L{\mathbb{d}r}} \right)}^{2}}i{\mathbb{d}r}\quad{\mathbb{d}t}}}$where$\frac{R{\mathbb{d}r}}{L{\mathbb{d}r}}\quad{is}\quad{the}\quad{inverse}\quad{of}\quad{the}\quad{Direct}\quad{Rotor}\quad{Time}\quad{{Constant}\left( {1\text{/}\sec} \right)}$and$R{\mathbb{d}{r\left( \frac{M\mathbb{d}}{L{\mathbb{d}r}} \right)}^{2}}{is}{\quad\quad}{the}\quad{Direct}\quad{Rotor}\quad{Excitation}\quad{Constant}\quad({ohms})$The vqr input to inverse transform block 402 is(R(iqr)+Lld(wre)idr+wre(λdr)) as implemented in the mathblock15 408.Mathblock16 416 implements R(iqr), the quadrature value of resistivevoltage drop in the stator 118. Mathblock17 418 implements Lld(wre)idr,the quadrature inductive back electromotive force in the stator.Mathblock18 420 implements wre(λdr), the quadrature air gap flux backelectromotive force resulting from the airgap 174 between the stator andthe rotor. The quadrature air gap flux term is evaluated as follows:${\lambda\quad{qr}} = {{\int_{0}^{t}{{- \frac{Rqr}{Lqr}}\quad\lambda\quad{qr}}} + {R\quad{{qr}\left( \frac{Mq}{Lqr} \right)}^{2}{iqr}\quad{\mathbb{d}t}}}$where$\frac{Rqr}{Lqr}\quad{is}\quad{the}\quad{inverse}\quad{of}\quad{the}\quad{Quadrature}\quad{Rotor}\quad{Time}\quad{{Constant}\left( {1\text{/}\sec} \right)}$and$R{\mathbb{d}{r\left( \frac{Mq}{Lqr} \right)}^{2}}{is}{\quad\quad}{the}\quad{Quadrature}\quad{Rotor}\quad{Excitation}\quad{Constant}\quad({ohms})$

FIG. 5 shows the bi-directional AC-to-DC electric power converter 500comprising converter 136. The converter includes a driver module 121that receives control voltages vd, vq from the first hybrid converter180 via control voltage bus 178. The driver module provides three pairsof outputs Da1/Da2, Db1/Db2, and Dc1/Dc2. The driver output pairs areconnected to the respective gates of semiconductor switches a1/a2,b1/b2, and c1/c2. Driver operation turns the switches on and off. Theemitters of semiconductors a1, b1, c1 are interconnected with thecollectors of semiconductors a2, b2, c2 and phases a, b, c at phasejunctions t1, t2, t3. The collectors of a1, b1, c1 are connected topositive DC link 190; the emitters of a2, b2, c2 are connected to anegative DC link 188. Appendix 1 provides additional details relating tothe feedforward controls for synchronous reluctance machines.

In operation, the feedforward controller for a synchronous reluctancemachine 100 controls the bi-directional exchange of mechanical powerbetween a mechanical energy exchange device 104 (like a flywheel) andthe synchronous reluctance motor-generator 102. The motor-generator maybe operated in generating modes and in charging modes. In the generatingmode, mechanical energy is transferred to the motor-generator when themotor-generator exerts a resisting torque T2 tending to slow therotational speed wre of the shaft 112.

During the generating mode, electric power is generated at a variablefrequency depending upon the speed of the shaft wre and the number ofpoles on the motor-generator rotor 114. The bi-directional AC-to-DCelectric power converter 136 receives electric power from themotor-generator via phase conductors a, b, c and provides a DC output onDC bus 146. Capacitor 186 provides both ripple control/smoothing of theoutput and electric energy storage. Electrical network 138 and thecapacitor are electrical loads when the motor-generator is generatingelectric power.

During the charging mode, mechanical energy is transferred to themechanical energy exchange device 104 when the motor-generator exerts anadvancing torque T1 tending to increase the rotational speed wre of theshaft 112. The bi-directional AC-to-DC electric power converter 136receives DC power from the electric network 138 via the DC bus 146,converts the DC power to AC power and transfers AC power to themotor-generator via phase conductors a, b, c.

The bi-directional AC-to-DC electric power converter 136 is controlledby first and second hybrid controllers 180, 182. The second hybridconverter provides a current setpoint Ipeak to the first hybridconverter: Ipeak is a function of DC bus current Ibus and voltage Vbus.The second hybrid controller provides control voltage outputs vd, vq;the control voltage outputs vd, vq are functions of Ipeak. Driver module121 synthesizes pulse width modulated (PWM) gate driver outputs 502 thatare a function of inputs vd, vq from the control voltage bus 178.Converter 136 exchanges electric power between the AC bus 134 and the DCbus 146 as semiconductor switches within the converter are modulated bythe driver module's PWM outputs 502.

The second hybrid converter's Ipeak output is the output of a SPDTswitch S5. The switch selects either an output of first controller 202or an output of the second controller 204. With reference to FIG. 6, thesecond hybrid converter can be operated in a charging mode or adischarging mode. In charging mode, Vbus is greater than Vcharge and S5is at setting 2; the second controller's output Ipeakc is selected. Incharging mode, the lesser of a current error (Ireg-Ibus) or a maximumcharge rate Ichargemax is input to second PI controller 216 whose outputis Ipeakc (See FIG. 2). The Ireg value is derived from the product of anamps/volt ratio zcharge and a voltage error (Vbus-Vcharge). As long as(Ireg-Ibus) remains below Ichargemax, the charge current will becontrolled by the amps/volt ratio zcharge. Otherwise, the maximum chargecurrent will be limited by Ichargemax. This charging profile may beadapted to duplicate that of an electric storage battery or anotherelectric energy storage device.

With continued reference to FIG. 6, the first controller 202 functionsas a feedback controller. In the first controller a constant operatingpoint of the machine is used to determine the direct and quadraturecurrents in the rotor reference frame for a peak current command Ipeak.A positive Ipeak command causes the motor-generator 102 to act as agenerator in generating mode while a negative Ipeak command causes themotor-generator to act as a motor in charging mode. When Vbus is lessthan or equal to Vcharge, the second hybrid controller 182 is indischarging mode and switch S5 is at setting 1; the first controller'soutput Ipeakb is selected. The error between the bus voltage and aregulation voltage (Vreg−Vbus) is input to first PI controller 208. Theoutput of the controller Irestore is combined with the load current Ibusto determine a total DC bus current command Isum. DC bus current Isum isin turn converted into a peak current command Ipeakb by mathblock1 210.This control regime maintains a minimum DC bus voltage of Vreg. Thisdischarging profile may be adapted to duplicate that of an electricstorage battery or another electric energy storage device.

The first hybrid controller 180 includes third controller 300, afeedback controller and a fourth controller 400, a feedforwardcontroller. Referring now to FIG. 7, the controller selected dependsupon the shaft speed wre. During acceleration of the shaft 112, thethird controller (feedback control) is used for acceleration phases A1and A2 having respective low speed (0 to wre1) and medium speed (>wre1to wre2) ranges. Acceleration A3 in the high speed range (>wre2 to wre3)is under the control of the fourth controller (feedforward control).Similarly, during full speed operation and initial deceleration of theshaft, the fourth controller (feedforward control) is used for fullspeed operation FS (wre3) and for deceleration speed range D1 (wre3 towre2). Deceleration phases D2 and D3 through respective medium (<wre2 towre1) and low (<wrel to de-minimus) speed ranges are under the controlof the third controller (feedback control).

Third controller 300 includes a third PI controller 302 that operates tominimize current errors (Idr-idr) and (Iqr-iqr). Stator currents ia andib are converted to direct and quadrature values id and iq before beingtransformed into rotor reference frame values idr, iqr by transformblock 304. Setpoint currents Idr and Idq are derived from Ipeak. The PIcontroller outputs vdr and vqr in the rotor reference frame aretransformed by inverse transform block 306 into command voltages vd, vq.When closed, DPST switch S3 interconnects the third controller outputsvd, vq to the command voltage bus 178.

Third controller 300 has operating modes depending upon rotor velocitywre. Switch S2 selects from Start-up angle, Zero, and Position inputs toprovide position signals to transform block 304 and inverse transformblock 306. When accelerating shaft 112 in the low speed range, Start-upis selected to provide a start-up rotor angle and fixed rotor velocitywre; this is an inductive start-up mode for the synchronous reluctancemotor-generator while charging. Upon reaching wre2, position is selectedand the actual rotor position is input. While decelerating and uponreaching wre1, Zero is selected to provide a “0” rotor position. Thisgenerates DC currents in the machine causing the rotor 107 to brakeslowing shaft 112 through a combination of induction and reluctancetorque.

First hybrid controller third controller 400 implements a model of thesynchronous reluctance motor-generator 102 to predict the controlvoltage commands vd, vq present on the control voltage bus 178. Directresistive, inductive, and air gap flux terms provide respective directvoltage terms 410, 412, 414 that are summed to predict the rotorreference frame direct control voltage vdr. Quadrature resistive,inductive, and air gap flux terms provide respective quadrature voltageterms 416, 418, 420 that are summed to predict the rotor reference framequadrature control voltage vqr. Inverse transform block 402 converts vdrto vd and inverse transform block 408 converts vqr to vq. Switch S4interconnects control voltages vd, vq with control voltage bus 178 whenthe S4 is closed. Table 1 below relates selected operating modes andfirst hybrid controller switch settings. TABLE 1 Modes and First HybridController Switch Settings Switch Switch Switch Mode S2 S3 S4 ControllerLow Speed Range Start Closed Open Feedback Accelerating Up Angle MediumSpeed Range Position Closed Open Feedback High Speed Range Position OpenClosed Feedforward Low Speed Range Zero Closed Open FeedbackDeceleratingControl voltage output commands vd, vq from the first hybrid controllerconnect with driver module 121 via control voltage bus 178. Drivermodule 121 provides pulse width modulated signals to sequentiallyoperate the semiconductor gates/switches of converter 136. Semiconductorswitching provides for the exchange of three phase electric power a, b,c between the AC bus 134 and the DC bus 146. Pulse width modulation ofthe semiconductor switches modulates the voltage of the AC and DC businterconnections and the quantity of electric power exchanged. Appendix1 provides additional details relating to the operation of thefeedforward controls for synchronous reluctance machines.

While various embodiments of the present invention have been describedabove, it should be understood that they have been presented by way ofexample only, and not limitation. It will be understood by those skilledin the art that various changes in form and details can be made thereinwithout departing from the spirit and scope of the invention as definedin the appended claims. Thus, the breadth and scope of the presentinvention should not be limited by any of the above-described exemplaryembodiments, but should be defined only in accordance with the followingclaims and their equivalents.

APPENDIX 1 FOLLOWS High-Speed Control of Synchronous Reluctance Machinewith Solid Rotor

1 Model of Solid-Rotor Synchronous Reluctance Machine

A solid rotor of a synchronous reluctance machine can be simply modelledin the rotor reference frame by direct and quadrature windings, similarto that of an induction machine model. The two-phaseflux-linkage/current relationships of the machine in the rotor referenceare then given by: $\begin{matrix}{{\begin{bmatrix}\lambda_{sd}^{r} \\\lambda_{rd}^{r}\end{bmatrix} = {\begin{bmatrix}L_{sd} & M_{d} \\M_{d} & L_{rd}\end{bmatrix}\begin{bmatrix}i_{sd}^{r} \\i_{rd}^{r}\end{bmatrix}}},} & (1) \\{\begin{bmatrix}\lambda_{sq}^{r} \\\lambda_{rq}^{r}\end{bmatrix} = {\begin{bmatrix}L_{sq} & M_{q} \\M_{q} & L_{rq}\end{bmatrix}\begin{bmatrix}i_{sq}^{r} \\i_{rq}^{r}\end{bmatrix}}} & (2)\end{matrix}$The stator voltage/current relationships are given by: $\begin{matrix}{\upsilon_{sd}^{r} = {{R_{s}i_{sd}^{r}} - {\omega_{re}\lambda_{sq}^{r}} + \frac{\mathbb{d}\lambda_{sd}^{r}}{\mathbb{d}t}}} & (3) \\{\upsilon_{sq}^{r} = {{R_{s}i_{sq}^{r}} - {\omega_{re}\lambda_{sd}^{r}} + \frac{\mathbb{d}\lambda_{sq}^{r}}{\mathbb{d}t}}} & (4)\end{matrix}$and the rotor dynamics are given by:$\frac{\mathbb{d}\lambda_{rd}^{r}}{\mathbb{d}t} = {{- R_{rd}}i_{rd}^{r}}$$\begin{matrix}{\frac{\mathbb{d}\lambda_{rq}^{r}}{\mathbb{d}t} = {{- R_{rq}}i_{rq}^{r}}} & (5)\end{matrix}$We will define the states of the system as the stator currents and therotor flux-linkages. The rotor currents can be rewritten as:$\begin{matrix}{{i_{rd}^{r} = {\frac{1}{L_{rd}}\left( {\lambda_{rd}^{r} - {M_{d}i_{sd}^{r}}} \right)}},} & (6) \\{{i_{rd}^{r} = {\frac{1}{L_{rd}}\left( {\lambda_{rd}^{r} - {M_{d}i_{sd}^{r}}} \right)}},} & (7)\end{matrix}$hence the rotor dynamics are given by: $\begin{matrix}{{\frac{\mathbb{d}\lambda_{rd}^{r}}{\mathbb{d}t} = {{{- \frac{R_{rd}}{L_{rd}}}\lambda_{rd}^{r}} + {R_{rd}\frac{M_{d}}{L_{rd}}i_{sd}^{r}}}},} & (8) \\{\frac{\mathbb{d}\lambda_{rq}^{r}}{\mathbb{d}t} = {{{- \frac{R_{rq}}{L_{rq}}}\lambda_{rq}^{r}} + {R_{rq}\frac{M_{q}}{L_{rq}}i_{sq}^{r}}}} & (9)\end{matrix}$and the stator fluxes can be written as $\begin{matrix}\begin{matrix}{\lambda_{sd}^{r} = {{L_{sd}i_{sd}^{r}} + {M_{d}i_{rd}^{r}}}} \\{= {{\frac{{L_{ad}L_{rd}} - M_{d}^{2}}{L_{rd}}i_{sd}^{r}} + {\frac{M_{d}}{L_{rd}}\lambda_{rd}^{r}}}} \\{{= {{L_{lad}i_{sd}^{r}} + {\frac{M_{d}}{L_{rd}}\lambda_{rd}^{r}}}},}\end{matrix} & (10) \\{\lambda_{sq}^{r} = {{L_{lsq}i_{sq}^{r}} + {\frac{M_{q}}{L_{rq}}\lambda_{rq}^{r}}}} & (11)\end{matrix}$and the dynamic equations for the stator are therefore given by:$\begin{matrix}\begin{matrix}{\upsilon_{sd}^{r} = {{R_{s}i_{sd}^{r}} - {\omega_{re}\left( {{L_{lsq}i_{sq}^{r}} + {\frac{M_{q}}{L_{rq}}\lambda_{rq}^{r}}} \right)} + {\frac{\mathbb{d}}{\mathbb{d}t}\left( {{L_{lsd}i_{sd}^{r}} + {\frac{M_{d}}{L_{rd}}\lambda_{rd}^{r}}} \right)}}} \\{= {{R_{s}i_{sd}^{r}} - {\omega_{re}\left( {{L_{lsq}i_{sq}^{r}} + {\frac{M_{q}}{L_{rq}}\lambda_{rq}^{r}}} \right)} + {L_{lsd}\frac{\mathbb{d}i_{sd}^{r}}{\mathbb{d}t}} + {\frac{M_{d}}{L_{rd}}\left( {{{- \frac{R_{rd}}{L_{rd}}}\lambda_{rd}^{r}} + {R_{rd}\frac{M_{d}}{L_{rd}}i_{sd}^{r}}} \right)}}} \\{{= {{\left( {R_{s} + {R_{rd}\frac{M_{d}^{2}}{L_{rd}^{2}}}} \right)i_{sd}^{r}} - {\omega_{re}L_{lsq}i_{sq}^{r}} - {\omega_{re}\frac{M_{q}}{L_{rq}}\lambda_{rq}^{r}} - {R_{rd}\frac{M_{d}}{L_{rd}^{2}}\lambda_{rd}^{r}} + {L_{lsd}\frac{\mathbb{d}i_{sd}^{r}}{\mathbb{d}t}}}},} \\{v_{sq}^{r} = {{R_{s}i_{sq}^{r}} - {\omega_{re}\left( {{L_{lsd}i_{sd}^{r}} + {\frac{M_{d}}{L_{rd}}\lambda_{rd}^{r}}} \right)} + {\frac{\mathbb{d}}{\mathbb{d}t}\left( {{L_{lsq}i_{sq}^{r}} + {\frac{M_{q}}{L_{rq}}\lambda_{rq}^{r}}} \right)}}} \\{= {{\left( {{Rs} + {R_{rq}\frac{M_{q}^{2}}{L_{rq}^{2}}}} \right)i_{sq}^{r}} - {\omega_{re}L_{lsd}i_{sd}^{r}} + {\omega_{re}\frac{M_{d}}{L_{rd}}\lambda_{rd}^{r}} - {\frac{R_{rq}M_{q}}{L_{rq}^{2}}\lambda_{rq}^{r}} + {L_{lsq}\frac{\mathbb{d}i_{sq}^{r}}{\mathbb{d}t}}}}\end{matrix} & (12)\end{matrix}$The complete dynamic equations for the system are therefore given by:$\begin{matrix}{{\frac{\mathbb{d}\lambda_{rd}^{r}}{\mathbb{d}t} = {{{- \frac{R_{rd}}{L_{rd}}}\lambda_{rd}^{r}} + {R_{rd}\frac{M_{d}}{L_{rd}}i_{sd}^{r}}}},} & (13) \\{\frac{\mathbb{d}\lambda_{rq}^{r}}{\mathbb{d}t} = {{{- \frac{R_{rq}}{L_{rq}}}\lambda_{rq}^{r}} + {R_{rq}\frac{M_{q}}{L_{rq}}i_{sq}^{r}}}} & (14) \\{\frac{\mathbb{d}i_{sd}^{r}}{\mathbb{d}t} = {\frac{1}{L_{lsd}}{\quad{\left\lbrack {\upsilon_{sd}^{r} - {\left( {R_{s} + {R_{rd}\frac{M_{d}^{2}}{L_{rd}^{2}}}} \right)i_{sd}^{r}} + {\omega_{re}L_{lsq}i_{sq}^{r}} + {\omega_{re}\frac{M_{q}}{L_{rq}}\lambda_{rq}^{r}} + {R_{rd}\frac{M_{d}}{L_{rd}^{2}}\lambda_{rd}^{r}}} \right\rbrack,}}}} & (15) \\{\frac{\mathbb{d}i_{sq}^{r}}{\mathbb{d}t} = {\frac{1}{L_{lsq}}{\quad\left\lbrack {\upsilon_{sq}^{r} - {\left( {R_{s} + {R_{rq}\frac{M_{q}^{2}}{L_{rq}^{2}}}} \right)i_{sq}^{r}} + {\omega_{re}L_{lsd}i_{sd}^{r}} - {\omega_{re}\frac{M_{d}}{L_{rd}}\lambda_{rd}^{r}} + {\frac{R_{rq}M_{q}}{L_{rq}^{2}}\lambda_{rq}^{r}}} \right\rbrack}}} & (16)\end{matrix}$By defining a new flux variable, $\begin{matrix}{{\lambda_{ad}^{r} = {\frac{M_{d}}{L_{rd}}\lambda_{rd}^{r}}},} & (17) \\{\lambda_{aq}^{r} = {\frac{M_{q}}{L_{rq}}\lambda_{rq}^{r}}} & \quad\end{matrix}$The dynamics can then be rewritten as follows, in vector format:$\begin{matrix}{{\frac{\mathbb{d}{\overset{\rightarrow}{\lambda}}_{a}^{r}}{\mathbb{d}t} = {{{- \left\lbrack \frac{R_{r}}{L_{r}} \right\rbrack}{\overset{\rightarrow}{\lambda}}_{a}^{r}} + {\left\lbrack {R_{r}\left( \frac{M}{L_{r}} \right)}^{2} \right\rbrack{\overset{\rightarrow}{i}}_{s}^{r}}}},{\frac{\mathbb{d}{\overset{\rightarrow}{i}}_{a}^{r}}{\mathbb{d}t} = {\left\lbrack L_{la} \right\rbrack^{- 1}\left\{ {{\overset{\rightarrow}{\upsilon}}_{s}^{r} - {\left\lbrack {R_{s} + {R_{r}\left( \frac{M}{L_{r}} \right)}^{2}} \right\rbrack{\overset{\rightarrow}{i}}_{s}^{r}} + {{j\omega}_{re}\left( {{{L_{la}}{\overset{\rightarrow}{i}}_{s}^{r}} + {\overset{\rightarrow}{\lambda}}_{a}^{r}} \right)} + {\left\lbrack \frac{R_{r}}{L_{r}} \right\rbrack{\overset{\rightarrow}{\lambda}}_{a}^{r}}} \right\}}},} & (18)\end{matrix}$where the [x] represent diagonal matrices with direct and quadratureparameters along the diagonal. The dynamics can then be expressed interms of 4 sets of parameters, rotor time constants$\left\lbrack \frac{R_{r}}{L_{r}} \right\rbrack,$rotor excitation constants$\left\lbrack {R_{r}\left( \frac{M}{L_{r}} \right)}^{2} \right\rbrack,$stator leakage inductances [L_(ts)], and the stator resistance R_(s).The stator leakage inductance can most likely be assumed to be a scalar,and this inductance and the stator resistance can be determined quicklythrough terminal measurements of the stator sans rotor.

The parameters$\left\lbrack \frac{R_{r}}{L_{r}} \right\rbrack\quad{{and}\quad\left\lbrack {R_{r}\left( \frac{M}{L_{r}} \right)}^{2} \right\rbrack}$can be determined as follows:

-   -   Command either a direct or quadrature current is i^(r) _(sx) to        the machine.    -   Instantaneously disable the PWM to the machine. The stator        current should quickly (ideally instantaneously) go to zero.

In this case the stator voltage generated by the machine will be due tothe rotor flux: $\begin{matrix}\begin{matrix}{{\overset{\rightarrow}{\upsilon}}_{s}^{r} = {{{j\omega}_{re}\left\lbrack \frac{M}{L_{r}} \right\rbrack}{\overset{\rightarrow}{\lambda}}_{r}^{r}}} \\{= {{j\omega}_{re}{\overset{\rightarrow}{\lambda}}_{a}^{r}}}\end{matrix} & (19)\end{matrix}$From this voltage we can therefore easily determine the flux linkage{right arrow over (X)}^(r) _(a). From the exponential decays of thevoltage waveforms we can determine the rotor time constants$\left\lbrack \frac{R_{r}}{L_{r}} \right\rbrack.$This can best be done through a curve fitting of the extracted data.From the initial conditions we can determine the excitation parameters,for both direct and quadrature, as follows: $\begin{matrix}{{\left( \frac{M_{x}^{2}}{L_{rx}} \right) = \frac{\lambda_{ax}^{r}\left( {t = 0} \right)}{i_{ax}}},{{R_{rx}\left( \frac{M_{x}}{L_{rx}} \right)}^{2} = {\left( \frac{R_{rs}}{L_{rx}} \right)\left( \frac{M_{x}^{2}}{L_{rx}} \right)}},} & (20)\end{matrix}$The stator leakage inductances L_(tsd) and L_(tsq) can be assumed to beequal, and can be determined by measuring the inductance of the statorwindings with the rotor removed.2 Control Technique2.1 Feedforward Control

The control algorithm currently in use at high speed is a feedforwardmethod. Because of the nature of the flywheel system, it isstraightforward to model the machine dynamics accurately. Hence, we canuse the model developed above to determine the appropriate commandvoltages applied to the machine. The steady-state voltages for desiredcurrents i_(sd) ^(r) and i_(sq) ^(r) and resulting air-gap fluxes λ_(ad)^(r) and λ_(aq) ^(r) are given as follows: $\begin{matrix}{{\upsilon_{sd}^{r} = {{\left\lbrack {R_{s} + {R_{rd}\left( \frac{M_{d}}{L_{rd}} \right)}^{2}} \right\rbrack i_{sd}^{r}} - {\omega_{re}L_{lsq}i_{sq}^{r}} - {\omega_{re}\lambda_{aq}^{r}} - {R_{rd}\frac{M_{d}}{L_{rd}^{2}}\lambda_{ad}^{r}}}},{\upsilon_{sq}^{r} = {{\left\lbrack {R_{s} + {R_{rq}\left( \frac{M_{q}}{L_{rq}} \right)}^{2}} \right\rbrack i_{sq}^{r}} - {\omega_{re}L_{lsd}i_{sd}^{r}} + {\omega_{re}\lambda_{ad}^{r}} - {R_{rq}\quad\frac{M_{q}}{L_{rq}^{2}}\lambda_{aq}^{r}}}}} & (21)\end{matrix}$The terms${R_{rx}\left( \frac{M_{q}}{L_{rq}} \right)}^{2}i_{sx}^{r}{\quad\quad}{and}\quad R_{rx}\quad\frac{M_{x}}{L_{rx}^{2}}\lambda_{ax}^{r}$will cancel each other once the “air-gap” flux reaches its final value.Because of this, and because of the difficulties associated with thedetermination of the $R_{rx}\frac{M_{x}}{L_{rx}^{2}}$parameter, and because these terms are relativelysmall compared to the other terms at high-speed, we approximate thesteady-state relations as follows: $\begin{matrix}{{v_{sd}^{r} \approx {{R_{s}i_{sd}^{r}} - {\omega_{re}L_{1{sq}}i_{sq}^{r}} - {\omega_{re}\lambda_{aq}^{r}}}},{v_{sq}^{r} \approx {{R_{s}i_{sq}^{r}} + {\omega_{re}L_{1{sd}}i_{sd}^{r}} - {\omega_{re}\lambda_{ad}^{r}}}},} & (22)\end{matrix}$The “air-gap” flux can be estimated from the desired stator currents asfollows: $\begin{matrix}{{\lambda_{ad}^{r} = {{\int_{0}^{t}{{- \frac{R_{rd}}{L_{rd}}}\lambda_{ad}^{r}}} + {{R_{rd}\left( \frac{M_{d}}{L_{rd}} \right)}^{2}i_{sd}^{r}\quad{\mathbb{d}t}}}},{\lambda_{aq}^{r} = {{\int_{0}^{t}{{- \frac{R_{rq}}{L_{rq}}}\lambda_{aq}^{r}}} + {{R_{rq}\left( \frac{M_{q}}{L_{rq}} \right)}^{2}i_{sq}^{r}\quad{\mathbb{d}t}}}}} & (23)\end{matrix}$2.2 Deadtime CompensationAs part of the feedforward control, we will also need to compensate forthe deadtime effect. The dead-time associated with the phase legs willalter the desired average-value output voltage of the phase as follows:$\begin{matrix}{{\left\langle {v_{out}(t)} \right\rangle = {\left\langle {v_{command}(t)} \right\rangle - {\frac{V_{bus}t_{d}}{T_{s}}\frac{i_{out}}{{i_{out}(t)}}}}},} & (24)\end{matrix}$where t_(d) is the “dead” time and T_(s) is the switching period, andV_(bus) is the bus voltage. We can compensate for the fundamentalcomponent of the deadtime voltage using the command currents as follows:$\begin{matrix}{{v_{cd}^{r} = {v_{sd}^{r} + {\frac{4V_{bus}t_{d}}{\pi\quad T_{s}}\frac{i_{sd}^{r}}{i_{pk}}}}},{v_{cq}^{r} = {v_{sq}^{r} + {\frac{4V_{bus}t_{d}}{\pi\quad T_{s}}\frac{{\mathbb{i}}_{sd}^{r}}{i_{pk}}}}}} & (25)\end{matrix}$2.3 Concurrent Stationary ControlBecause of the high electrical frequencies, the average-value outputs ofthe phase legs of the inverter will include small DC and sub-synchronouscomponents. These can be reduced with a concurrent stationary PIfeedback controller. As the bandwidth of this controller is centeredaround DC, a low bandwidth regulator will not interfere with thehigh-frequency desired currents.

Bus Voltage Regulator

The purpose of the flywheel system is to provide bus voltage regulationshould loss of external power occur. The flywheel system comprises aninverter interconnecting a DC bus and an electric motor and a load ornetwork interconnected to the inverter by the DC bus. The systemincludes a DC bus capacitor C to sustain the bus voltage during anoutage until the flywheel system can power up. The current exchangedwith the DC bus by the inverter is Iflywheel and the current exchangedwith the DC bus by the load or network is Ibus. Summation of currents atthe capacitor terminals yields the following expression: $\begin{matrix}{{I_{bus} + {C\frac{\mathbb{d}V_{bus}}{\mathbb{d}t}}} = I_{flywheel}} & (1)\end{matrix}$Assuming 100% efficiency of the flywheel system, the current from theflywheel system is determined through conservation of power:$\begin{matrix}{I_{flywheel} = \frac{{\tau\omega}_{r}}{V_{bus}}} & (2)\end{matrix}$where τ is the electromagnetic torque applied to the flywheel shaft andω_(r) is the rotor velocity. An expression for the bus voltage dynamicsis therefore given by $\begin{matrix}{\frac{\mathbb{d}V_{bus}}{\mathbb{d}t} = {\frac{1}{C}\left( {\frac{{\tau\omega}_{r}}{V_{bus}} - I_{bus}} \right)}} & (3)\end{matrix}$We therefore regulate the bus voltage through regulation of torque onthe flywheel system. To linearize the dynamics and provide a feedbackcontroller with a feedforward term for the bus current, the followingcommand torque is used: $\begin{matrix}{{\overset{\_}{\tau} = {\frac{v_{bus}}{\omega_{r}}\left( {I_{bus} + {K_{p}e_{v}} + {K_{i}{\int_{0}^{t}{e_{v}\quad{\mathbb{d}t}}}} + {K_{d}\frac{\mathbb{d}e_{v}}{\mathbb{d}t}}} \right)}},{where}} & (4) \\{e_{v} = {{\overset{\_}{V}}_{1{bus}} - V_{bus}}} & (5)\end{matrix}$The gains K_(p), K_(i), and K_(d) are chosen to optimize response time.The torque command is then used to determine a peak current command tothe motor. Under steady-state conditions, the flywheel torque is givenby: $\begin{matrix}{\tau = {\frac{3P}{4}\left( {L_{d} - L_{q}} \right)i_{d}^{r}i_{q}^{r}}} & (6)\end{matrix}$The direct and quadrature currents in the rotor reference frame arereferred to the peak motor current I_(pk) as follows:i _(d) ^(r) =K _(d) I _(pk)  (7)i _(q) ^(r) =K _(q) I _(pk)  (8)whereK _(q)=√{square root over (1−K _(d) ²)}  (9)The peak current command is therefore given by: $\begin{matrix}{\quad{{\overset{\_}{I}}_{pk} = \sqrt{\frac{4\overset{\_}{\tau}}{3{P\left( {L_{d} - L_{q}} \right)}K_{d}\sqrt{1 - K_{d}^{2}}}}}} & (10)\end{matrix}$

1. A synchronous reluctance machine and control system comprising: abi-directional AC-to-DC electric power converter electricallyinterconnecting and bi-directionally exchanging electric power between asynchronous reluctance motor-generator and a DC bus wherein said powerexchange is controlled by a plurality of current controllers operablycoupled to said converter wherein at least one of said controllers is afeedforward controller.
 2. The energy conversion system of claim 1wherein the motor-generator includes an electrical stator spaced apartfrom an all-metal rotor by an air gap.
 3. The energy conversion systemof claim 2 wherein a rotatable coupling interconnects andbi-directionally exchanges mechanical power between the rotor and aflywheel mass.
 4. The energy conversion system of claim 3 wherein the atleast one feedforward controller evaluates a plurality ofmotor-generator dependant control voltage terms and sums selected termsto produce a control voltage output operably coupled to the converter.5. The energy conversion system of claim 4 wherein the control voltageoutput is a function of any one or combination of stator resistivevoltage drop, stator inductive back electromotive force, and air-gapflux back electromotive force.
 6. The energy conversion system of claim5 wherein at least one of the controllers is a feedback controller thatlimits the error between a stator current and a first current setpoint.7. The energy conversion system of claim 6 wherein the at least onefeedforward controller and the at least one feedback controller producetwo control voltage outputs representing direct and quadrature voltages.8. The energy conversion system of claim 7 wherein the at least onefeedback controller controls the power exchange during a flywheel lowspeed range and the at least one feedforward controller controls thepower exchange during a flywheel high speed range.
 9. An energyconversion system comprising: a bi-directional AC-to-DC electric powerconverter electrically interconnecting and bi-directionally exchangingelectric power between a synchronous reluctance motor-generator havingan all-metal rotor and a DC bus, the power exchange being controlled bya plurality of current controllers operably coupled to the converterwherein a first controller is a feedforward controller, a secondcontroller is a feedback controller and a control voltage output of atleast one controller is a function of peak current.